## 1. Introduction

Recent studies of observations (Feldstein and Lee 1998; Hartmann and Lo 1998), general circulation models (Feldstein and Lee 1996), and mechanistic dynamical models (Robinson 1991, 1996; Yu and Hartmann 1993;Lee and Feldstein 1996) appear to be converging upon a dynamical understanding of variations in the zonal index.^{1} Specifically,

anomalies in the zonally averaged zonal wind often persist for 10 or more days;

such persistence is due, at least in part, to the reinforcement of anomalies in the zonally averaged zonal flow by transient eddy momentum fluxes; this is denoted positive eddy feedback; and

only eddies with relatively high frequencies, periods of 10 or fewer days, contribute to the positive eddy feedback. Eddies with longer timescales either drive variations in the zonal index stochastically or, in some models, contribute systematically to their dissipation.

The dynamical basis of positive eddy feedback remains unclear. Hartmann (1995) and Hartmann and Zuercher (1998) argue that positive eddy feedback derives from the sensitive dependence of baroclinic life cycles on the preexisting meridional shear. While their life cycle calculations are convincing, it is not clear how to carry the results of initial-value life cycle experiments over to the real atmosphere where baroclinic eddies are always present. An alternative suggestion was made by Robinson (1996), who argued that the baroclinic component of the zonal index was crucial for positive eddy feedback. Robinson based his argument on the meridional coincidence, found in a mechanistic model, between anomalously strong poleward eddy heat fluxes and anomalously strong westerlies.

The proposed feedback loop is as follows: through the action of surface drag a region of anomalously strong barotropic westerlies becomes a region of enhanced baroclinicity. Baroclinic eddies are generated more vigorously in such a region, and as the eddies propagate meridionally away from their latitudes of generation, their momentum fluxes reinforce the anomalously strong barotropic westerlies, thereby closing the loop. It should be emphasized that this argument, and its implications for the relative timing of increases in heat and momentum fluxes, applies only to that portion of the variability in eddy activity that is organized by variations in the large-scale flow. Clearly, much of the variability in eddy activity is effectively stochastic. This is true even in mechanistic models (e.g., Robinson and Qin 1992).

Here we attempt to quantify under what conditions a baroclinic positive eddy feedback can work. The acceleration of barotropic westerlies in a region of baroclinic eddy generation over the course of a nonlinear baroclinic life cycle is an established result (e.g., Simmons and Hoskins 1978), but for this feedback loop to function, the zonally averaged low-level baroclinicity in this region must be reinforced in the response of the mean flow to eddy transports of heat and momentum. A key result of this paper is that this is indeed possible, but only through the action of surface drag.

We also attempt to explain some more subtle aspects of the zonal index that have emerged from observations and models:

the poleward drift of anomalies in the zonally averaged zonal wind (Feldstein 1998);

the meridionally banded structure of anomalies in both the zonal wind and its eddy forcing (Robinson 1996; Feldstein and Lee 1998);

in a mechanistic model, the dependence of positive eddy feedback on surface drag (Robinson 1996); and

in a mechanistic model (Robinson 1994, 1996), the frequency dependence of the positive eddy feedback.

The last two points need elucidation. While it is impossible to unequivocally demonstrate the existence of positive feedback in observations, and it is very difficult to do so in a full GCM, simple models can be manipulated in such a way as to verify the existence of and to quantify the strength of feedbacks operating within these models. Robinson (1994) did this by inserting an unphysical zonally symmetric source of zonal momentum into the equations of motion of a fully nonlinear, two-level, global, primitive equations model. The source of momentum was barotropic and was dipolar in latitude, thus mimicking the structure of zonal momentum variations generated internally by the model. The source oscillated in time, and experiments were run using a range of oscillation frequencies. It was found that transient eddy feedbacks significantly modified the response of the zonal winds to the external forcing and that this modification acted increasingly as a positive feedback as the period of the oscillation was increased. The eddy feedback was negative for periods less than 30 days. A simple linear positive feedback—that is, a negative drag—does not behave in this manner.

Robinson (1996) performed control simulations using the same two-level model and varying the strength of surface drag. The model was then modified by fixing the barotropic component of the zonal wind to the time average from the appropriate control run, the model was run with the barotropic zonal flow held fixed, and the eddy momentum fluxes were saved. Using these fluxes, the variability of the zonal flow that would have resulted *if* it had been permitted was calculated. For strong surface drag the zonal wind variations so diagnosed were far weaker, especially at low frequencies, than those generated in the control simulation. For weak surface drag, however, the diagnosed variations in the zonal flow were similar to those normally produced by the model. These experiments confirmed the presence of a positive eddy feedback in the model, operating primarily at low frequencies—periods of 30 or more days—and only in the presence of relatively strong surface drag.

In this paper we demonstrate that the above described features of the zonal index may be understood as straightforward consequences of the quasigeostrophic dynamics of the zonally averaged flow in the presence of surface drag and interior radiative damping, conditional on some plausible, but as yet unproven, assumptions about the behavior of transient baroclinic eddies:

that baroclinic eddies are generated more vigorously in regions of stronger low-level baroclinicity, and

that baroclinic eddies typically propagate away from their latitude of generation before they dissipate, and because of the sphericity of the earth, this propagation is more often equatorward than poleward.

The next section lays out the theoretical argument, and a few numerically calculated examples are shown in section 3. These results can be viewed as a tropospheric application of the principle of “downward control” (Haynes et al. 1991). Section 4 discusses the application of the results to understanding the zonal index.

## 2. Theory

### a. Basic equations

*ζ*is the relative vorticity,

*σ*is a static stability parameter, and Φ is the geopotential. Taking the meridional derivative of (1) and using (2) and the geostrophic relation, we obtain an equation for the evolution of the zonally averaged zonal wind driven by the eddy fluxes of potential vorticity,

*t*indicates differentiation with respect to time.

*p*=

*p*

_{b}) is derived from the zonally averaged thermodynamic equation. Defining, following Bretherton (1966), potential vorticity in a sheet at the lower boundary as

*p*=

*p*

_{b}for (3),

*τ*

_{N}is the radiative dissipation time. When this is included in (3), it becomes

*p*=

*p*

_{b},

*τ*

_{E}, is typically about five days. At

*p*= 0 the pressure vertical velocity vanishes. If it is assumed that the eddy heat flux also vanishes, the temperature cannot change, and

Because the zonally averaged problem is linear in the zonally averaged flow, these equations apply equally to the time-averaged zonal flow and to anomalies. The emphasis here is on understanding the zonal index, so that the latter interpretation is most relevant. We may, therefore, think of all the quantities in this paper—eddy fluxes and zonal mean responses—as deviations from some climatological mean.

### b. The transient limit

*p*=

*p*

_{b},

### c. The steady limit

*y,*but there is no such differentiation on the left-hand side, so regions of strong eddy forcing are accompanied on their north and south flanks by lobes of oppositely signed responses in the zonal wind. From (14) it can be seen that poleward eddy heat fluxes at the surface need not be accompanied by decreased baroclinicity if there is a sufficiently strong westerly maximum with respect to latitude in the surface zonal wind. Then the increase with latitude in adiabatic cooling by the Ekman-induced vertical velocity can compensate for the poleward eddy heat flux.

### d. Response to boundary forcing

^{2}For a barotropic solution, the lower-boundary condition (9) becomes

### e. Forcing aloft

*υ**

*q**]. Enhanced low-level baroclinicity [negative left-hand side of (17)] in a region of baroclinic wave generation can, therefore, be maintained in steady state only if the bulk of the eddy activity propagates to neighboring latitudes before dissipating. Specifically, the generation region must coincide with the wings of the distribution of equatorward (negative) eddy flux of potential vorticity, where its meridional second derivative is negative.

### f. Time dependence

It is simplest to think about the time dependence of the response in the case of surface forcing alone and further, to assume, reasonably, that the Ekman damping time *τ*_{E} is less than the radiative damping time *τ*_{N}. Consider the response to a surface poleward eddy flux turned on at some time. The barotropic flow adjusts to its final value (16) over a time proportional to *τ*_{E}. During this adjustment, the poleward temperature gradient at the surface increases. After a time on the order of *τ*_{E}, the right-hand side and the final term on the left-hand side of (9) come into approximate balance, so that the magnitude of the strongest poleward surface temperature gradient (reduction in baroclinicity) scales approximately with *τ*_{E}. Thereupon, the surface temperature gradient decays with an *e*-folding time equal to *τ*_{N}.

## 3. Examples

For the zonal flow problem, the distribution of the eddy flux of potential vorticity is constrained only by (7). Reasonable choices can, however, be made by reference to the theory of baroclinic instability and to observations. In the time average there will be a region of positive [*υ***q**] associated with the irreversible poleward transport of heat at the surface by baroclinic eddies. There must be compensating regions of equatorward fluxes of potential vorticity elsewhere in the atmosphere. In baroclinic life cycle calculations baroclinic eddies propagate to the upper troposphere and about 15° of latitude equatorward before depositing their wave activity [e.g., Edmon et al.’s (1980) diagnosis of the Simmons and Hoskins (1978) experiment]. Thus, we choose distributions of [*υ***q**] that are positive in some region at the lower boundary and have a compensating negative region in the upper troposphere.

*p*

_{b}= 1000 mb, and

*p*

_{tr}= 200 mb. Amplitudes here and below are chosen arbitrarily to give responses on the order of 10 in the appropriate units. From the results of Hartmann and Lo it appears that typical anomalous eddy forcing associated with the zonal index is on the order of 10 m s

^{−1}day

^{−1}. Numerical solutions are obtained using successive overrelaxation on a 101 × 101 grid, extending meridionally from −7500 to 7500 km and from 1000 to 0 mb. The Coriolis parameter is set to a typical midlatitude value,

*f*

_{0}= 10

^{−4}s

^{−1}. A constant lapse rate of 6 K km

^{−1}is used between 1000 and 200 mb, with a surface temperature of 288 K. The lapse rate is zero for pressures less than 200 mb. Second-order finite differences are used, except at the boundaries where one-sided differences are taken.

The transient response to this boundary forcing is shown in Fig. 1. There is an acceleration of the zonal flow, and decreasing low-level baroclinicity in the center of the domain is flanked by regions of increasing baroclinicity. The steady response to this same forcing (not shown) is, as described above, purely barotropic.

*y*−

*y*

_{0}| >

*w*/

*y*−

*y*

_{0}| = (3/2)

^{1/2}

*w.*Note that increased baroclinicity occurs in two regions: the region of baroclinic eddy generation poleward of where the eddies are dissipated, and in a region equatorward of the dissipation region.

Figures 2a,b show the steady-state results with the parameters *A*_{b} = 10^{−5} m s^{−2} × (*p*_{b} − *p*_{tr}) *w* = 1500 km, *τ*_{E} = 6.5 days, where *p*_{l} = 500 mb, *p*_{tr} = 200 mb, and the two values of the offset are *y*_{0}, = 0 and −1500 km. The superposition of the baroclinic response to the interior forcing, which shifts with the value of *y*_{0}, and the barotropic response to the surface forcing, centered at *y* = 0, is clearly evident.

Figure 3 shows the temporal adjustment of the surface baroclinicity at *y* = 0 for the same forcing parameters used to obtain Fig. 2b. The curves, obtained using different values of the dissipation parameters, have the characteristic shape of the difference between two decaying exponentials. The dependence on *τ*_{E} is shown in Fig. 3a. Consistent with (17), the asymptotic value of the baroclinicity at long times is independent of *τ*_{E}. The strength of the early reduction in baroclinicity and, therefore, the time required to approach the steady-state solution scale approximately linearly with *τ*_{E}. The radiative damping time (Fig. 3b) has less influence on the strongest negative baroclinicity. For larger values of *τ*_{N}, the approach to steady state is slower, but the final enhancement of the baroclinicity is greater.

## 4. Discussion

These results demonstrate that the mean flow modification by baroclinic eddies may indeed reinforce the low-level baroclinicity in their latitudes of generation, if the eddies propagate away before they dissipate. Figure 4 shows a schematic of this process. Baroclinic eddies propagate upward and equatorward, whereupon they break and are dissipated, resulting in a convergence of the Eliassen–Palm flux. This much of the picture is identical to that obtained for inviscid baroclinic life cycles (Edmon et al. 1980). In inviscid life cycle calculations, however, the thermal gradient is permanently expelled from the source latitudes of the baroclinic eddies. Here, the presence of surface friction allows a steady state to be achieved, wherein warming by the sinking branch of the transformed Eulerian mean circulation (Holton 1992, chapter 10) enhances the equatorward thermal gradient across the source latitudes.

If baroclinic eddies are generated more vigorously in regions of stronger low-level baroclinicity, this result provides a mechanism for the positive eddy feedback on zonal flow variations in midlatitudes. Furthermore, the present results are consistent with the features of zonal flow variations that were noted in the introduction.

Poleward drift of zonal flow anomalies. If, due to the sphericity of the earth, eddy activity predominantly propagates equatorward from its region of generation, then the low-level baroclinicity is reinforced poleward of where the eddies are generated. Thus, there is a poleward shift of the positive eddy feedback; and the anomalous low-level baroclinicity, vertically integrated zonal wind, and region of eddy generation will tend to drift poleward together. This poleward drift is limited, presumably, to latitudes where the climatological background baroclinicity is strong.

Meridional bandedness of the zonal wind and eddy forcing. In the steady state, the zonal flow response and the surface baroclinicity depend on the second meridional derivative of the eddy forcing. Thus, a single maximum in the westward eddy forcing gives rise to a deceleration of the zonal flow flanked by regions of acceleration, and a single maximum in the eddy forcing leads to multiple regions of strengthened baroclinicity.

Dependence of positive eddy feedback on surface drag. The steady response of the surface baroclinicity to a given eddy forcing is independent of the strength of the surface drag. In the presence of stronger surface drag, however, the approach to this steady state is accelerated, bringing the system more rapidly into a state of positive eddy feedback.

Frequency dependence of eddy feedback. Low-level baroclinicity in the region of eddy generation is reduced immediately after the eddy forcing is turned on but is increased in the steady state. The zonal flow response at lower frequencies will more closely resemble the steady-state response, while that at higher frequencies will resemble the transient behavior. Thus, we expect a negative eddy feedback at high frequencies and a positive feedback at low frequencies, which is consistent with what was found in a mechanistic model (Robinson 1994). This is confirmed by solutions obtained (not shown) for the temporal Fourier transforms of (8), (9), and (10).

Despite these promising elements of agreement with observations and models, the theory presented here is clearly incomplete in that the distribution of eddy forcing has simply been assumed, and only zonally symmetric dynamics have been considered. Furthermore, these results lead to positive eddy feedback only if it is true that baroclinic eddies are generated more vigorously at latitudes of stronger than usual low-level baroclinicity. In the absence of a complete theory that allows us to calculate eddy quantities in terms of the mean flow,^{3} the necessary next step is a careful analysis of the observed relationships between eddy fluxes and zonally averaged fields.

## Acknowledgments

The author appreciates the constructive comments from three anonymous reviewers and gratefully acknowledges the support of this work by the National Science Foundation, Grant ATM-9628850.

## REFERENCES

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Steady response of the zonal winds to combined boundary and interior forcing. The forcing parameters are given in the text. The contour interval is 1 m s^{−1} and negative contours are dashed. The region of interior eddy forcing is shaded. (a) *y*_{0} = 0 and (b) *y*_{0} = −1500 km.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

Steady response of the zonal winds to combined boundary and interior forcing. The forcing parameters are given in the text. The contour interval is 1 m s^{−1} and negative contours are dashed. The region of interior eddy forcing is shaded. (a) *y*_{0} = 0 and (b) *y*_{0} = −1500 km.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

Steady response of the zonal winds to combined boundary and interior forcing. The forcing parameters are given in the text. The contour interval is 1 m s^{−1} and negative contours are dashed. The region of interior eddy forcing is shaded. (a) *y*_{0} = 0 and (b) *y*_{0} = −1500 km.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

Time-dependent response of the surface baroclinicity at *y* = 0. (a) With varied values of the Ekman dissipation time *τ*_{E}. (b) With varied values of the radiative dissipation time *τ*_{N}.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

Time-dependent response of the surface baroclinicity at *y* = 0. (a) With varied values of the Ekman dissipation time *τ*_{E}. (b) With varied values of the radiative dissipation time *τ*_{N}.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

Time-dependent response of the surface baroclinicity at *y* = 0. (a) With varied values of the Ekman dissipation time *τ*_{E}. (b) With varied values of the radiative dissipation time *τ*_{N}.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

A schematic (described in the text) of the process that leads to the enhancement of baroclinicity in the source region of baroclinic eddies.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

A schematic (described in the text) of the process that leads to the enhancement of baroclinicity in the source region of baroclinic eddies.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

A schematic (described in the text) of the process that leads to the enhancement of baroclinicity in the source region of baroclinic eddies.

Citation: Journal of the Atmospheric Sciences 57, 3; 10.1175/1520-0469(2000)057<0415:ABMFTE>2.0.CO;2

^{1}

Here the zonal index can be defined as the latitude of the zonally averaged jet. Variations in the zonal index represent fluctuations in the central latitude of the jet.

^{2}

Note that this is true in any region above *all* of the eddy forcing. This is another example of the “principle of downward control.” Because mechanical dissipation is confined to the planetary boundary layer, secondary circulations generated by eddy forcing must close downward, and there is no vertical motion to generate baroclinicity in the layer above the top of the forcing.

^{3}

Whitaker and Sardeshmukh’s (1998) storm track model is promising in this regard, though such models have not been particularly successful in treating the response to zonally symmetric variations in the mean flow.